# Reflection in a plane.

What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part of its properties anymore.

My aim is to prove that this type of reflection is linear and that $R(u).R(u)=u.u$, can you help me? It seems so obvious that I can't actually prove it...

Thank you

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Where did the problem come from? Does the source tell you what a reflection's definition is? – rschwieb May 23 '14 at 19:38

Given a dot product and a nonzero vector $a,$ the reflection $\tau_a$ is given by $$\tau_a (x) = x - \left( \frac{2 a \cdot x}{a \cdot a}\right) a$$ Straightforward to calculate $\tau_a (x) \cdot \tau_a(x).$

Note that a reflection, along with being an isometry, is also self adjoint, meaning its matrix will be symmetric as well as orthogonal; we write this as $$\tau_a (x) \cdot y = x \cdot \tau_a(y)$$If this seems peculiar (it does to me), consider that, given the reflection matrix $R,$ there is some orthogonal matrix $P,$ meaning $P^T P = I,$ such that $P^T R P$ is a diagonal matrix with $n-1$ diagonal entries equal to $1$ and the final diagonal entry equal to $-1.$

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and strightforward that it's linear :) – rschwieb May 23 '14 at 19:37
Yes, I see how to prove from there... What does $(\frac{2a.x}{a.a})$ represent? – Kika May 23 '14 at 19:37
I'm guessing the distance from the point to the plane but I don't see how, can you clarify that? – Kika May 23 '14 at 19:38
@Timmy, why don't you see what happens to different vectors $x,$ once when $x=a$ and once when $x$ is perpendicular to $a$. – Will Jagy May 23 '14 at 19:39
Oh yes, I see it now, thank you! – Kika May 23 '14 at 19:41

It is the unique isometry of $R^3$ whose fixed point set is exactly the plane $a\cdot r=0$.

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How can I prove my aims from there? – Kika May 23 '14 at 19:32
Starting from this definition, one would have to prove that an isometry fixing a plane is linear, which would take a little work... – rschwieb May 23 '14 at 19:36
Exactly! But the question was what is the definition as opposed to properties. I gave the definition. Everything else are the properties. It is a nice exercise in geometry to show that every isometry of 3-space fixing the origin is in fact linear. – studiosus May 23 '14 at 19:41
@studiosus Sure. it just makes me despair a bit to get questions like this. It somewhat presumes there is only one definition, and makes one wonder how the student could be given question without the definition. – rschwieb May 23 '14 at 19:43
@rchwieb: agree. My guess is that the definition is in the textbook, but the student did not bother reading some earlier chapters. – studiosus May 23 '14 at 19:57